URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) WEAK SOLUTIONS FOR A VISCOUS p-LAPLACIAN EQUATION CHANGCHUN LIU Abstract

ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

WEAK SOLUTIONS FOR A VISCOUS p-LAPLACIAN EQUATION

CHANGCHUN LIU

Abstract. In this paper, we consider the pseudo-parabolic equation ut− k∆ut = div(|∇u|^p−2∇u). By using the time-discrete method, we establish the existence of weak solutions, and also discuss the uniqueness.

1. Introduction

This paper concerns the study of the viscousp-Laplacian equation

∂u

∂t −k∂∆u

∂t = div(|∇u|^p−2∇u), x∈Ω, p >2, (1.1) with boundary condition

u

_∂Ω= 0, (1.2)

and initial condition

u(x,0) =u0(x), x∈Ω. (1.3)

Here Ω is a bounded domain in R^N and k > 0 is the viscosity coefficient. The termk^∂∆u_∂t in (1.1) is interpreted as due to viscous relaxation effects, or viscosity;

hence, the equation (1.1) is called “viscousp-Laplacian equations”. The well-known p-Laplacian equation is obtained by settingk= 0.

Equation (1.1) arises as a regularization of the pseudo-parabolic equation

∂u

∂t −k∂∆u

∂t = ∆u, (1.4)

which arises in various physical phenomena. (1.4) can be assumed as a model for diffusion of fluids in fractured porous media [1, 5, 4], or as a model for heat conduction involving a thermodynamic temperatureθ=u−k∆uand a conductive temperature u[10, 3]. Equation (1.4) has been extensively studied, and there are many outstanding results concerning existence, uniqueness, regularity, and special properties of solutions, see for example [4, 5, 6, 7, 8, 9, 11].

To derive (1.4), B. D. Coleman, R. J. Duffin and V. J. Mizel considered a special kinematical situation, of nonsteady simple shearing flow [4]. In fact, when the influence of many factors, such as the molecular and ion effects, are considered, one has the nonlinear relation div(|∇u|^p−2∇u) in stead of ∆uin right-hand side of (1.4). Hence, we obtain (1.1).

2000Mathematics Subject Classification. 35G25, 35Q99, 35K55, 35K70.

Key words and phrases. Pseudo-parabolic equations, existence, uniqueness.

c

2003 Southwest Texas State University.

Submitted August 5, 2002. Published June 10, 2003.

1

Equation (1.1) is something like the p-Laplacian equation, but many methods which are useful for thep-Laplacian equation are no longer valid for this equation.

Because of the degeneracy, problem (1.1)-(1.3) does not admit classical solutions in general. So, we study weak solutions in the sense of following

DefinitionA functionuis said to be a weak solution of (1.1)-(1.3), if the following conditions are satisfied:

(1) u∈L^∞(0, T;W₀^1,p(Ω))∩C(0, T;H¹(Ω)), ^∂u_∂t ∈L^∞(0, T;W^−1,p0(Ω)), where p⁰ is conjugate exponent ofp.

(2) For ϕ∈C₀^∞(QT) andQT = Ω×(0, T), Z Z

Q_T

u∂ϕ

∂tdx dt+k Z Z

Q_T

∇u∂∇ϕ

∂t dx dt− Z Z

Q_T

|∇u|^p−2∇u∇ϕdx dt= 0. (3) u(x,0) =u0(x).

In this paper, we discuss first the existence of weak solutions. Most proofs of existence for (1.4) are based on the Yoshida approximations [6], but these methods do not apply to (1.1). Our method for proving the existence of weak solutions is based on a time discrete method that constructs approximate solutions. Later on, we discuss the uniqueness of a solution. For simplicity we setk= 1 in this paper.

2. Existence of weak solutions

Theorem 2.1. If u0 ∈W₀^1,p(Ω) with p >2, then problem (1.1)-(1.3) has at least one solution.

We use the a discrete method for constructing an approximate solution. First, divide the interval (0, T) in N equal segments and seth= _N^T. Then consider the problem

1

h(uk+1−uk)−1

h(∆uk+1−∆uk) = div(|∇uk+1|^p−2∇uk+1), (2.1) uk+1|∂Ω= 0, k= 0,1, . . . , N−1, (2.2) whereu₀is the initial value.

Lemma 2.2. For a fixed k, if uk ∈ H₀¹(Ω), problem (2.1)-(2.2) admits a weak solution uk+1∈W₀^1,p(Ω), such that for anyϕ∈C₀^∞(Ω), have

1 h

Z

Ω

(uk+1−uk)ϕdx+1 h

Z

Ω

(∇uk+1− ∇uk)∇ϕdx+ Z

Ω

|∇uk+1|^p−2∇uk+1∇ϕdx= 0.

(2.3) Proof. On the spaceW₀^1,p(Ω), we consider the functionals

Φ1[u] = 1 p

Z

Ω

|∇u|^pdx,

Φ2[u] = 1 2

Z

Ω

|u|²dx,

Φ3[u] = 1 2

Z

Ω

|∇u|²dx,

Ψ[u] = Φ1[u] + 1

hΦ2[u] + 1

hΦ3[u]− Z

Ω

f udx,

where f ∈ H⁻¹(Ω) is a known function. Using Young’s inequality, there exist constantsC1, C2>0, such that

Ψ[u] = 1 p

Z

Ω

|∇u|^pdx+ 1 2h

Z

Ω

|u|²dx+ 1 2h

Z

Ω

|∇u|²dx− Z

Ω

f u dx

≥C1

Z

Ω

|∇u|^pdx−C2kfk−1;

hence Ψ[u]→ ∞, askuk1,p→+∞. Herekuk1,pdenotes the norm ofuinW₀^1,p(Ω).

Since the norm is lower semi-continuous andR

Ωf udxis a continuous functional, Ψ[u] is weakly lower semi-continuous on W₀^1,p(Ω) and satisfying the coercive con- dition. From [2] we conclude that there existsu_∗∈W₀^1,p(Ω), such that

Ψ[u_∗] = inf Ψ[u],

andu∗is the weak solutions of the Euler equation corresponding to Ψ[u], 1

hu− 1

h∆u−div(|∇u|^p−2∇u) =f.

Taking f = (u_k −∆u_k)/h, we obtain a weak solutions u_k+1 of (2.1)–(2.2). The

proof is complete.

Now, we need to establish a priori estimates, for the weak solutions u_k+1 of (2.1)–(2.2). First, we define the weak solutions of (1.1)–(1.3) as follows:

u^h(x, t) =u_k(x), kh < t≤(k+ 1)h, k= 0,1, . . . , N−1, u^h(x,0) =u0(x).

Lemma 2.3. The weak solutionsuk of (2.1)–(2.2) satisfy

h

N

X

k=1

Z

Ω

|∇uk|^pdx≤C, (2.4)

sup

0<t<T

Z

Ω

|∇u^h(x, t)|^pdx≤C, (2.5)

whereC is a constant independent ofhandk.

Proof. i) We takeϕ=uk+1 in the integral equality (2.3) (we can easily prove that forϕ∈W₀^1,p(Ω), (2.3) also holds).

1 h

Z

Ω

(u_k+1−u_k)u_k+1dx+1 h

Z

Ω

(∇uk+1− ∇uk)∇uk+1dx +

Z

Ω

|∇uk+1|^p−2∇uk+1∇uk+1dx= 0, i.e.,

1 h

Z

Ω

|uk+1|²dx+1 h

Z

Ω

|∇uk+1|²dx−1 h

Z

Ω

ukuk+1dx

−1 h

Z

Ω

∇uk+1∇ukdx+ Z

Ω

|∇uk+1|^pdx= 0.

Thus

1 h

Z

Ω

|u_k+1|²dx+ 1 h

Z

Ω

|∇u_k+1|²dx+ Z

Ω

|∇u_k+1|^pdx

= 1 h

Z

Ω

ukuk+1dx+1 h

Z

Ω

∇uk+1∇ukdx.

By Young’s inequality, 1

h Z

Ω

|uk+1|²dx+ 1 h

Z

Ω

|∇uk+1|²dx+ Z

Ω

|∇uk+1|^pdx

≤ 1 2h

Z

Ω

|uk|²dx+ 1 2h

Z

Ω

|uk+1|²dx+ 1 2h

Z

Ω

|∇uk|²dx+ 1 2h

Z

Ω

|∇uk+1|²dx;

that is,

1 2

Z

Ω

|uk+1|²dx+1 2 Z

Ω

|∇uk+1|²dx+h Z

Ω

|∇uk+1|^pdx

≤1 2

Z

Ω

|uk|²dx+1 2 Z

Ω

|∇uk|²dx.

(2.6)

Adding these inequalities fork from 0 toN−1, we have h

N

X

k=1

Z

Ω

|∇u_k|^pdx≤ 1 2

Z

Ω

|u₀|²dx+1 2

Z

Ω

|∇u₀|²dx.

Therefore, (2.4) holds.

ii) We takeϕ=uk+1−uk in the integral equality (2.3) and have 1

h Z

Ω

(u_k+1−u_k)(u_k+1−u_k)dx+ 1 h

Z

Ω

(∇uk+1− ∇uk)∇(uk+1−u_k)dx +

Z

Ω

|∇u_k+1|^p−2∇u_k+1∇(u_k+1−u_k)dx= 0. Since the first term and the second term of the left hand side of the above equality is nonnegative, it follows that

Z

Ω

|∇uk+1|^pdx≤ Z

Ω

|∇uk+1|^p−2∇uk+1∇ukdx

≤p−1 p

Z

Ω

|∇uk+1|^pdx+1 p

Z

Ω

|∇uk|^pdx;

thus,

Z

Ω

|∇uk+1|^pdx≤ Z

Ω

|∇uk|^pdx.

For anym, with 1≤m≤N−1, adding the above inequality forkfrom 0 tom−1, we have

Z

Ω

|∇u_m|^pdx≤ Z

Ω

|∇u₀|^pdx.

Therefore, (2.5) holds.

Lemma 2.4. For a weak solutions uk+1 of (2.1)–(2.2), we have

−Ch≤ Z

Ω

|uk+1|²dx+ Z

Ω

|∇uk+1|²dx− Z

Ω

|uk|²dx− Z

Ω

|∇uk|²dx≤0, (2.7) whereC is a constant independently ofh.

Proof. The second inequality in (2.7) is an immediate consequence of (2.6). To prove the first inequality, we chooseϕ=uk in (2.3) and obtain

1 h

Z

Ω

(uk+1−uk)ukdx+ 1 h

Z

Ω

(∇uk+1− ∇uk)∇ukdx +

Z

Ω

|∇uk+1|^p−2∇uk+1∇ukdx= 0. Therefore,

Z

Ω

|uk|²dx+ Z

Ω

|∇uk|²dx− Z

Ω

u_k+1u_kdx− Z

Ω

∇uk+1∇ukdx

=h Z

Ω

|∇uk+1|^p−2∇uk+1∇ukdx

≤hZ

Ω

|∇u_k+1|^pdx(p−1)/pZ

Ω

|∇u_k|^pdx1/p

.

Here we have used H¨older inequality. By (2.5) again, we obtain Z

Ω

|uk|²dx+ Z

Ω

|∇uk|²dx− Z

Ω

uk+1ukdx− Z

Ω

∇uk+1∇ukdx≤Ch.

Therefore, Z

Ω

|uk|²dx+ Z

Ω

|∇uk|²dx

≤Ch+ Z

Ω

uk+1ukdx+ Z

Ω

∇uk+1∇ukdx

≤Ch+1 2

Z

Ω

|uk+1|²dx+1 2 Z

Ω

|uk|²dx+1 2 Z

Ω

|∇uk+1|²dx+1 2

Z

Ω

|∇uk|²dx.

i.e., Z

Ω

|uk|²dx+ Z

Ω

|∇uk|²dx− Z

Ω

|uk+1|²dx− Z

Ω

|∇uk+1|²dx≤Ch

which completes the proof.

Lemma 2.5.

sup

0<t<T

Z

Ω

|u^h|²dx+ Z

Ω

|∇u^h|²dx

≤ Z

Ω

|u0|²dx+ Z

Ω

|∇u0|²dx. (2.8) The proof follows by adding (2.4), for mwith 1≤m≤N −1, fork from 0 to m−1.

Proof of Theorem 2.1. First, we define the operatorA^t, A^t(∇u^h) =|∇uk|^p−2∇uk,

∆^hu^h=uk+1−uk, wherekh < t≤(k+ 1)h, k= 0,1, . . . , N−1. By the dispersion equation (2.1) and the (2.4) in Lemma 2.2, we know that

1

h(u_k+1−u_k) inL^∞(0, T;W^−1,p0(Ω)) is bounded. (2.9)

By (2.5), (2.7), (2.9) and (2.4) we known that exists a subsequence of{u^h}(which we denote as the original sequence) such that

u^h→u inL^∞(0, T;W^1,p(Ω)) weak-?,

∇u^h→ ∇u inL^∞(0, T;L²(Ω)) weak-?, 1

h(u_k+1−u_k)→ ∂u

∂t in L^∞(0, T;W^−1,p0(Ω) weak-?,

|∇u^h|^p−2∇u^h→w in L^∞(0, T;L^p0(Ω)) weak-?,

wherep⁰ is conjugate exponent ofp. From (2.3), we known, for anyϕ∈C₀^∞(QT), Z Z

Q_T

(1

h∆^hu^hϕ+1

h∆^h∇u^h∇ϕ+|∇u^h|^p−2∇u^h∇ϕ)dx dt= 0, i.e.,

Z Z

QT

(1

h∆^hu^hϕ− 1

h∆^hu^h∆ϕ+|∇u^h|^p−2∇u^h∇ϕ)dx dt= 0.

Lettingh→0, we obtain, in the sense of distributions,

∂u

∂t −∂∆u

∂t −div(w) = 0. (2.10)

Next, we prove thatw=|∇u|^p−2∇ua.e. inQT. Define fh(t) = t−kh

2h Z

Ω

|uk+1|²dx+ Z

Ω

|∇uk+1|²dx− Z

Ω

|uk|²dx− Z

Ω

|∇uk|²dx

+1 2

Z

Ω

|uk|²dx+1 2

Z

Ω

|∇uk|²dx,

wherekh < t≤(k+ 1)h. by (2.7) we have 1

2 Z

Ω

|∇uk|²dx+1 2 Z

Ω

|uk|²dx−Ch≤fh(t)≤ 1 2

Z

Ω

|uk|²dx+1 2

Z

Ω

|∇uk|²dx,

−C≤f_h⁰(t)≤0.

By Ascoli–Arzela theorem, there exists a functionf(t)∈C([0, T]), such that

h→0limfh(t) =f(t) fort∈[0, T] uniformly.

Using (2.7), we have

h→0lim 1

2 Z

Ω

|∇u^h|²dx+1 2

Z

Ω

|u^h|²dx

=f(t) fort∈[0, T] uniformly. (2.11) By (2.6) again, we obtain

1 2

Z

Ω

|uN|²dx+1 2 Z

Ω

|∇uN|²dx+ ZZ

Q_T

|∇u^h|^pdx dt≤ 1 2

Z

Ω

|u0|²dx+1 2

Z

Ω

|∇u0|²dx.

In the above inequality lettingh→0, and using (2.10) we have

h→0lim Z Z

Q_T

|∇u^h|^pdx dt≤f(0)−f(T)

= lim

ε→0

1 ε

Z T−ε

0

(f(t)−f(t+ε))dt

= lim

ε→0lim

h→0

h 1 2ε

Z T−ε

0

Z

Ω

(|u^h(x, t)|²− |u^h(x, t+ε)|²)dx dt + 1

2ε Z T−ε

0

Z

Ω

(|∇u^h(x, t)|²− |∇u^h(x, t+ε)|²)dx dti .

Since Φ2[u] = ¹₂R

Ω|u|²dxand Φ3[u] = ¹₂R

Ω|∇u|²dx are convex functionals, and δΦ2[u]

δu =u, δΦ3[u]

δu =−∆u, we have

1 2

Z

Ω

|u^h(x, t)|²dx−1 2 Z

Ω

|u^h(x, t+ε)|²dx +1

2 Z

Ω

|∇u^h(x, t)|²dx−1 2

Z

Ω

|∇u^h(x, t+ε)|²dx

≤ Z

Ω

u^h(x, t)(u^h(x, t)−u^h(x, t+ε))dx +

Z

Ω

∇u^h(x, t)(∇u^h(x, t)− ∇u^h(x, t+ε))dx.

Therefore,

h→0lim 1 2ε

hZ T−ε

0

Z

Ω

|u^h(x, t)|²− |u^h(x, t+ε)|²)dx dt +

Z T−ε

0

Z

Ω

(|∇u^h(x, t)|²− |∇u^h(x, t+ε)|²)dx dti

≤ 1 ε

Z T−ε

0

Z

Ω

(u(x, t)−u(x, t+ε))u dx dt +1

ε Z T−ε

0

Z

Ω

(∇u(x, t)− ∇u(x, t+ε))∇u dx dt . Hence, we obtain

lim

h→0

Z Z

Q_T

|∇u^h|^pdx dt≤ − Z T

0

h∂u

∂t, uidt+ Z T

0

h∂u

∂t,∆uidt, whereh,idenotes the inner product. Form (2.10), we obtain

h→0lim Z Z

Q_T

|∇u^h|^pdx dt≤ Z Z

Q_T

w∇udx dt . (2.12)

Again by ^δΦ_δu^1[u] = −div(|∇u|^p−2∇u) and the convexity of Φ1[u], for any g ∈ L^∞(0, T;W₀^1,p(Ω)) we have

−1 p

Z Z

Q_T

|∇g|^pdx dt+1 p

Z Z

Q_T

|∇u^h|^pdx dt

≤ Z Z

Q_T

−div(|∇u^h|^p−2∇u^h)(u^h−g)dx dt, that is

1 p

Z Z

QT

|∇g|^pdx dt−1 p

Z Z

QT

|∇u^h|^pdx dt≥ Z Z

QT

div(|∇u^h|^p−2∇u^h)(u^h−g)dx dt

= Z Z

QT

(|∇u^h|^p−2∇u^h)∇(g−u^h)dx dt.

By (2.11) andF(u) is weakly lower semicontinuous, in above equality lettingh→0, we obtain

1 p

Z Z

Q_T

|∇g|^pdx dt−1 p

Z Z

Q_T

|∇u|^pdx dt≤ Z Z

Q_T

w∇(g−u)dx dt. (2.13) In (2.13), we takeg=εg+uto obtain

1 ε

h1 p

Z Z

Q_T

|∇(εg+u)|^pdx dt−1 p

Z Z

Q_T

|∇u|^pdx dti

≥ Z Z

Q_T

w∇g dx dt.

Lettingε→0, Z Z

Q_T

δΦ1[u]

δu g dx dt= Z Z

Q_T

|∇u|^p−2∇u∇g dx dt≥ Z Z

Q_T

w∇g dx dt .

Sinceg is arbitrary, takingg=−g, we get the opposite inequality above; hence w=|∇u|^p−2∇u.

The strong convergence ofu^hin C(0, T;H¹(Ω)) and the fact thatu^h(x,0) =u₀(x)

completes the proof.

3. Uniqueness of solutions

In this section, we prove that the weak solution is unique. To this end we need the following lemma.

Lemma 3.1. For ϕ ∈ L^∞(t1, t2;W₀^1,p(Ω)) with ϕt ∈ L²(t1, t2;H¹(Ω)), the weak solutions uof the problem (1.1)-(1.3) onQT satisfies

Z

Ω

u(x, t1)ϕ(x, t1)dx+ Z

Ω

∇u(x, t1)∇ϕ(x, t1)dx +

Z t2

t₁

Z

Ω

u∂ϕ

∂t +∇u∂∇ϕ

∂t − |∇u|^p−2∇u∇ϕ

dx dt

= Z

Ω

u(x, t2)ϕ(x, t2)dx+ Z

Ω

∇u(x, t2)∇ϕ(x, t2)dx.

In particular, forϕ∈W₀^1,p(Ω), we have Z

Ω

(u(x, t1)−u(x, t2))ϕdx+ Z

Ω

∇(u(x, t1)−u(x, t2))∇ϕdx

− Z t2

t₁

Z

Ω

|∇u|^p−2∇u∇ϕdx dt= 0.

(3.1)

Proof. From ϕ ∈ L^∞(t1, t2;W₀^1,p(Ω)) and ϕt ∈ L²(t1, t2;H¹(Ω)), it follows that there exists a sequence of functions {ϕk}, for fixed t ∈(t1, t2), ϕk(·, t)∈ C₀^∞(Ω), and ask→ ∞

kϕkt−ϕtk_L2(t1,t2;H¹(Ω))→0, kϕk−ϕk_L∞(t1,t2;W₀^1,p(Ω))→0.

Choose a function j(s) ∈ C₀^∞(R) such that j(s) ≥ 0, for s ∈ R; j(s) = 0, for

∀|s|>1;R

Rj(s)ds= 1. Forh >0, definejh(s) =_h¹j(_h^s) and ηh(t) =

Z t−t1−2h

t−t2+2h

jh(s)ds.

Clearlyηh(t)∈C₀^∞(t1, t2), lim_h→0+ηh(t) = 1, for allt∈(t1, t2). In the definition of weak solutions, chooseϕ=ϕk(x, t)ηh(t). We have

Z t2

t1

Z

Ω

uϕkjh(t−t1−2h)dx dt− Z t2

t1

Z

Ω

uϕkjh(t−t2+ 2h)dx dt +

Z t2

t₁

Z

Ω

∇u∇ϕ_kj_h(t−t₁−2h)dx dt− Z t2

t₁

Z

Ω

∇u∇ϕ_kj_h(t−t₂+ 2h)dx dt +

Z t2

t₁

Z

Ω

uϕ_ktη_hdx dt+ Z t2

t₁

Z

Ω

∇u∇ϕktη_hdx dt

− Z t₂

t₁

Z

Ω

|∇u|^p−2∇u∇ϕkηhdx dt= 0.

Observe that

Z t₂

t₁

Z

Ω

uϕ_kj_h(t−t₁−2h)dx dt− Z

Ω

(uϕ_k)|t=t1dx

=

Z t₁+3h

t₁+h

Z

Ω

uϕ_kj_h(t−t₁−2h)dx dt−

Z t₁+3h

t₁+h

Z

Ω

(uϕ_k)|t=t2j_h(t−t₁−2h)dx dt

≤ sup

t1+h<t<t1+3h

Z

Ω

(uϕk)|t−(uϕk)|t₁

dx,

and u∈ C(0, T;L²(Ω)). We see that the right hand side tends to zero ash→0.

Similarly,

Z t₂

t1

Z

Ω

uϕkjh(t−t2+ 2h)dx dt− Z

Ω

(uϕk)|t=t₂dx

→0, ash→0,

Z t2

t1

Z

Ω

∇u∇ϕ_kj_h(t−t₁−2h)dx dt− Z

Ω

(∇u∇ϕ_k)|_t=t1dx

→0, ash→0,

Z t2

t₁

Z

Ω

∇u∇ϕ_kj_h(t−t₂+ 2h)dx dt− Z

Ω

(∇u∇ϕ_k)|_t=t2dx

→0, ash→0.

Lettingh→0 andk→ ∞, we obtain Z

Ω

u(x, t1)ϕ(x, t1)dx+ Z

Ω

∇u(x, t1)∇ϕ(x, t1)dx +

Z t₂

t1

Z

Ω

u∂ϕ

∂t +∇u∂∇ϕ

∂t − |∇u|^p−2∇u∇ϕ dx dt

= Z

Ω

u(x, t₂)ϕ(x, t₂)dx+ Z

Ω

∇u(x, t₂)∇ϕ(x, t₂)dx.

In particular forϕ∈W₀^1,p(Ω), we have Z

Ω

(u(x, t₁)−u(x, t₂))ϕdx+ Z

Ω

(∇u(x, t1)− ∇u(x, t2))∇ϕdx

− Z t2

t1

Z

Ω

|∇u|^p−2∇u∇ϕ dx dt= 0

which completes the proof.

For a fixed τ ∈ (0, T), set hsatisfying 0 < τ < τ +h < T. Letting t₁ = τ, t2=τ+h, then multiply (3.1) by _h¹, forϕ∈W₀^1,p(Ω), we obtain

Z

Ω

(u_h(x, τ))_τϕ(x)dx+ Z

Ω

((∇u)_h(x, τ))_τϕ(x)dx+ Z

Ω

(|∇u|^p−2∇u)_h(x, τ)∇ϕdx= 0, (3.2) where

uh(x, t) = (₁

h

Rt+h

t u(·, τ)dτ, t∈(0, T −h),

0, t > T−h.

Theorem 3.2. Problem (1.1)-(1.3) admits only one weak solution.

Proof. Supposeu₁, u₂ are two solutions of (1.1)-(1.3), then Z

Ω

(u1(x, τ)−u2(x, τ))hτϕ(x)dx+ Z

Ω

((∇u1− ∇u2)h(x, τ))τϕ(x)dx +

Z

Ω

(|∇u1|^p−2∇u1− |∇u2|^p−2∇u2)h(x, τ)∇ϕdx= 0.

For a fixedτ, we takeϕ(x) = [u₁−u₂]_h∈W₀^1,p(Ω), and hence Z

Ω

(u1(x, τ)−u2(x, τ))hτ(u1−u2)hdx +

Z

Ω

∇(u1(x, τ)−u2(x, τ))hτ∇(u1−u2)hdx

=− Z

Ω

[(|∇u1|^p−2∇u1− |∇u2|^p−2∇u2)h](x, τ)∇(u1−u2)hdx, i.e.,

Z

Ω

(u1(x, τ)−u2(x, τ))hτ(u1−u2)hdx +

Z

Ω

∇(u1(x, τ)−u2(x, τ))hτ∇(u1−u2)hdx

=− Z

Ω

[(|∇u1|^p−2∇u1− |∇u2|^p−2∇u2)h](x, τ)∇(u1−u2)hdx.

Integrating the above equality with respect toτ over (0, t), Z

Ω

|(u1−u₂)_h|²(x, t)dx+ Z

Ω

|∇(u1−u₂)_h|²(x, t)dx≤0, we haveR

Ω|(u1−u2)h|²dx= 0; therefore,u1=u2. Acknowledgment. The author would like to thank referee for his/her valuable suggestions and for providing the references E. Di Benedtto & M. Pierre [5] and E.

Di Benedetto & R. E. Showalter [6].

References

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Angew. Math. Phys., 19(1968), 614-627.

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Amer. Math. Soc., 324(1991), 331–351.

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Mech. Anal., 49(1972), 57-78.

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Math. Anal., 1(1970), 1-26.

[10] T. W. Ting,A cooling process according to two-temperature theory of heat conduction, J.

Math. Anal. Appl., 45(1974), 23-31.

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Changchun Liu

Department of Mathematics, Nanjing Normal University, Nanjing 210097, China Department of Mathematics, Jilin University, Changchun 130012, China

E-mail address:[email protected]